reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th64:
  for p being Point of TOP-REAL 2 st p`2 >= 0 for x,a being Real
, r being positive Real st 0 <= a & a < 1 & |.p-|[x,r*a]|.| > r*
  a holds +(x,r).p > a
proof
  let p be Point of TOP-REAL 2 such that
A1: p`2 >= 0;
  set p1 = p`1, p2 = p`2;
  reconsider p2 as non negative Real by A1;
  let x,a be Real;
  let r be positive Real;
  assume that
A2: 0 <= a and
A3: a < 1;
  reconsider a9 = a as non negative Real by A2;
  reconsider ra = r*a as Real;
  assume
A4: |.p-|[x,r*a]|.| > r*a;
  |.|[x,0]|-|[x,r*a]|.| = |.|[x,r*a]|-|[x,0]|.| by TOPRNS_1:27
    .= |.|[x-x,ra-0]|.| by EUCLID:62
    .= |.ra.| by TOPREAL6:23
    .= r*a9 by ABSVALUE:def 1;
  then
A5: p1 <> x or p2 <> 0 by A4,EUCLID:53;
A6: p = |[p`1,p`2]| by EUCLID:53;
  then reconsider z = p as Element of Niemytzki-plane by A1,Lm1,Th18;
A7: +(x,r).z in the carrier of I[01];
  per cases by A2;
  suppose
A8: a = 0;
    then p <> |[x,r*0]| by A4,TOPRNS_1:28;
    then +(x,r).p <> 0 by A1,Th60;
    hence thesis by A7,A8,BORSUK_1:40,XXREAL_1:1;
  end;
  suppose
    +(x,r).p = 1;
    hence thesis by A3;
  end;
  suppose
A9: a > 0 & +(x,r).z <> 1;
A10: |[p1-x,p2-0]|`1 = p1-x by EUCLID:52;
A11: |[p1-x,p2]|`2 = p2 by EUCLID:52;
    p2 is non negative;
    then
A12: p in Ball(|[x,r]|,r) by A6,A5,A9,Def5;
    then
A13: +(x,r).p = |.|[x,0]|-p.|^2/(2*r*p2) by A6,Def5
      .= |.p-|[x,0]|.|^2/(2*r*p2) by TOPRNS_1:27
      .= |.|[p1-x,p2-0]|.|^2/(2*r*p2) by A6,EUCLID:62
      .= ((p1-x)^2+p2^2)/(2*r*p2) by A10,A11,JGRAPH_1:29;
    |.p-|[x,r*a]|.|^2 > (r*a)^2 by A2,A4,SQUARE_1:16;
    then
A14: |.|[p1-x,p2-r*a]|.|^2 > (r*a)^2 by A6,EUCLID:62;
A15: |[p1-x,p2-r*a]|`2 = p2-r*a by EUCLID:52;
    |[p1-x,p2-r*a]|`1 = p1-x by EUCLID:52;
    then (p1-x)^2+(p2-r*a)^2 > (r*a)^2 by A14,A15,JGRAPH_1:29;
    then (p1-x)^2+p2^2-2*p2*(r*a)+(r*a)^2 > (r*a)^2;
    then (p1-x)^2+p2^2-2*p2*r*a > 0 by XREAL_1:32;
    then
A16: (p1-x)^2+p2^2 > 2*p2*r*a by XREAL_1:47;
A17: p2 = 0 implies p in y=0-line by A6;
    Ball(|[x,r]|,r) misses y=0-line by Th21;
    then reconsider p2 as positive Real by A12,A17,XBOOLE_0:3;
A18: a*((2*p2*r)/(2*r*p2)) = a*1 by XCMPLX_1:60;
    +(x,r).p > (2*p2*r*a)/(2*r*p2) by A13,A16,XREAL_1:74;
    hence thesis by A18,XCMPLX_1:74;
  end;
end;
